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ExpToTrig[expr]

converts exponentials in expr to trigonometric functions.

Details
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Examples  
Basic Examples  
Scope  
Applications  
Properties & Relations  
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ExpToTrig

ExpToTrig[expr]

converts exponentials in expr to trigonometric functions.

Details

  • ExpToTrig generates both circular and hyperbolic functions.
  • ExpToTrig tries when possible to give results that do not involve explicit complex numbers.
  • ExpToTrig automatically threads over lists, as well as equations, inequalities and logic functions.

Examples

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Basic Examples  (2)

Convert from exponentials to trigonometric functions:

Wolfram Language code: ExpToTrig[Exp[I x]]

Convert from exponentials to hyperbolic functions:

Wolfram Language code: ExpToTrig[Exp[x]]

Scope  (6)

Convert from exponentials to trigonometric functions:

Wolfram Language code: ExpToTrig[2 ^ (I x)]

Convert from exponentials to hyperbolic functions:

Wolfram Language code: ExpToTrig[3 ^ x - 3 ^ -x]

Convert from logarithms to inverse trigonometric functions:

Wolfram Language code: ExpToTrig[Log[1 + I x] - Log[1 - I x]]

Convert from logarithms to inverse hyperbolic functions:

Wolfram Language code: ExpToTrig[Log[2 + x] - Log[2 - x]]

ExpToTrig converts rational powers of to the equivalent trigonometric expressions:

Wolfram Language code: ExpToTrig[(-1) ^ (5 / 7)]

ExpToTrig threads elementwise over lists, equations, inequalities and Boolean operators:

Wolfram Language code: ExpToTrig[{Exp[I * x] == -1, (-1) ^ (1 / 11)}]
Wolfram Language code: ExpToTrig[E ^ x ≤ 11 && 3 ^ (I x) == 7]

Applications  (2)

Show that the unit circle maps to an interval in the Joukowski map:

Wolfram Language code: z + 1 / z /. z -> Exp[I θ]//ExpToTrig

Find the hyperbolic forms of solutions to differential equations:

Wolfram Language code: DSolve[{y''[t] - y[t] == 0}, y[t], t]
Wolfram Language code: ExpToTrig[%]

Properties & Relations  (3)

ExpToTrig converts rational powers of to the equivalent trigonometric expressions:

Wolfram Language code: ExpToTrig[(-1) ^ (1 / 16)]

FunctionExpand represents the trigonometric expressions in terms of radicals:

Wolfram Language code: FunctionExpand[%]

ExpToTrig is the inverse of TrigToExp:

Wolfram Language code: TrigToExp[Sin[x]]
Wolfram Language code: ExpToTrig[%]

ExpToTrig threads element‐wise over lists, equations, inequalities and logic functions:

Wolfram Language code: ExpToTrig[Exp[I * x] == -1]
Wolfram Research (1996), ExpToTrig, Wolfram Language function, https://reference.wolfram.com/language/ref/ExpToTrig.html (updated 2007).

Text

Wolfram Research (1996), ExpToTrig, Wolfram Language function, https://reference.wolfram.com/language/ref/ExpToTrig.html (updated 2007).

CMS

Wolfram Language. 1996. "ExpToTrig." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2007. https://reference.wolfram.com/language/ref/ExpToTrig.html.

APA

Wolfram Language. (1996). ExpToTrig. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ExpToTrig.html

BibTeX

@misc{reference.wolfram_2026_exptotrig, author="Wolfram Research", title="{ExpToTrig}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/ExpToTrig.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_exptotrig, organization={Wolfram Research}, title={ExpToTrig}, year={2007}, url={https://reference.wolfram.com/language/ref/ExpToTrig.html}, note=[Accessed: 13-June-2026]}